Lemma 63.10.2. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a ring. The functors $Rf_!$ constructed in Section 63.9 are bounded in the following sense: There exists an integer $N$ such that for $E \in D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or $E \in D(X_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion, we have
$H^ i(Rf_!(\tau _{\leq a}E) \to H^ i(Rf_!(E))$ is an isomorphism for $i \leq a$,
$H^ i(Rf_!(E)) \to H^ i(Rf_!(\tau _{\geq b - N}E))$ is an isomorphism for $i \geq b$,
if $H^ i(E) = 0$ for $i \not\in [a, b]$ for some $-\infty \leq a \leq b \leq \infty $, then $H^ i(Rf_!(E)) = 0$ for $i \not\in [a, b + N]$.
Proof.
Assume $\Lambda $ is torsion and consider the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$. By construction it suffices to prove this when $f$ is an open immersion and when $f$ is a proper morphism. For any open immersion $j : U \to X$ of schemes, the functor $j_! : D(U_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ is exact and hence the statement holds with $N = 0$ in this case. If $f$ is proper then $Rf_! = Rf_*$, i.e., it is a right derived functor. Hence the bound on the left by Derived Categories, Lemma 13.16.1. Moreover in this case $f_* : \textit{Mod}(X_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(Y_{\acute{e}tale}, \Lambda )$ has bounded cohomological dimension by Morphisms, Lemma 29.28.5 and Étale Cohomology, Lemma 59.92.2. Thus we conclude by Derived Categories, Lemma 13.32.2.
Next, assume $\Lambda $ is arbitrary and let us consider the functor $Rf_! : D^+_{tors}(X_{\acute{e}tale}, \Lambda ) \to D^+_{tors}(Y_{\acute{e}tale}, \Lambda )$. Again we immediately reduce to the case where $f$ is proper and $Rf_! = Rf_*$. Again part (1) is immediate. To show part (3) we can use induction on $b - a$, the distinguished triangles of trunctions, and Étale Cohomology, Lemma 59.92.2. Part (2) follows from (3). Details omitted.
$\square$
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